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Comments on the fractional parts of Pisot numbers. (English) Zbl 1363.11072

Authors’ abstract: Let \(L(\theta,\lambda)\) be the set of limit points of the fractional parts \(\{\lambda\theta^n\}\), \(n=0,1,2,\dots\), where \(\theta\) is a Pisot number and \(\lambda\in \mathbb{Q}(\theta)\). Using a description of \(L(\theta,\lambda)\), due to Dubickas, we show that there is a sequence \((\lambda_n)_{n\geq 0}\) of elements of \(\mathbb{Q}(\theta)\) such that \(\text{Card}(L(\theta,\lambda_n)) < \text{Card}(L(\theta,\lambda_{n+1})),\) \(\forall n\geq 0.\) Also, we prove that the fractional parts of Pisot numbers, with a fixed degree greater than \(1,\) are dense in the unit interval.

MSC:

11J71 Distribution modulo one
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11R04 Algebraic numbers; rings of algebraic integers
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